Description

In the analysis of designed experiments the first stage is to compute the basic analysis of variance table, the estimate of the error variance (the residual or error mean square), σ̂2σ^2, and the (variance ratio) FF-statistic for the tt treatments. If this FF-test is significant then the second stage of the analysis is to explore which treatments are significantly different.

If there is a structure to the treatments then this may lead to hypotheses that can be defined before the analysis and tested using linear contrasts. For example, if the treatments were three different fixed temperatures, say 1818, 2020 and 2222, and an uncontrolled temperature (denoted by NN) then the following contrasts might be of interest.

18

20

22

N

(a)

(1/3)

(1/3)

(1/3)

− 1

(b)

− 1

0

1

0

182022N(a)131313-1(b)-1010

The first represents the average difference between the controlled temperatures and the uncontrolled temperature. The second represents the linear effect of an increasing fixed temperature.

For a randomized complete block design or a completely randomized design, let the treatment means be τ̂iτ^i, i = 1,2, … ,ti=1,2,…,t, and let the jjth contrast be defined by λijλij, i = 1,2, … ,ti=1,2,…,t, then the estimate of the contrast is simply:

t

Λj =

∑

τ̂iλij

i = 1

Λj=∑i=1tτ^iλij

and the sum of squares for the contrast is:

SSj = (Λj2)/( ∑ i = 1tλij2 / ni)

SSj=Λj2∑i=1tλij2/ni

(1)

where nini is the number of observations for the iith treatment. Such a contrast has one degree of freedom so that the appropriate FF-statistic is SSj / σ̂2SSj/σ^2.

The two contrasts λijλij and λij′λij′ are orthogonal if ∑ i = 1tλijλij′ = 0∑i=1tλijλij′=0 and the contrast λijλij is orthogonal to the overall mean if ∑ i = 1tλij = 0∑i=1tλij=0. In practice these sums will be tested against a small quantity, εε. If each of a set of contrasts is orthogonal to the mean and they are all mutually orthogonal then the contrasts provide a partition of the treatment sum of squares into independent components. Hence the resulting FF-tests are independent.

If the treatments come from a design in which treatments are not orthogonal to blocks then the sum of squares for a contrast is given by:

SSj = (ΛjΛj * )/( ∑ i = 1tλij2 / ni)

SSj=ΛjΛj*∑i=1tλij2/ni

(2)

where

t

Λj * =

∑

τi * λij

i = 1

Λj*=∑i=1tτi*λij

with τi * τi*, for i = 1,2, … ,ti=1,2,…,t, being adjusted treatment means computed by first eliminating blocks then computing the treatment means from the block adjusted observations without taking into account the non-orthogonality between treatments and blocks. For further details see John (1987) and Morgan (1993).

The rows of the analysis of variance table for the contrasts. For each row column 1 contains the degrees of freedom, column 2 contains the sum of squares, column 3 contains the mean square, column 4 the FF-statistic and column 5 the significance level for the contrast. Note that the degrees of freedom are always one and so the mean square equals the sum of squares.